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Heights and Mordell--Weil Group






This section pertains to height functions on the Mordell--Weil group 
of the Jacobian of genus two curves over the rationals.  The canonical
height is computed using the algorithm of Flynn and Smart[FS97]
 with improvements by Stoll[Sto99].



HeightConstant(J: parameters) : JacHyp -> FldPrElt, FldPrElt



    Effort: RngIntElt                   Default: 0



    Factor: BoolElt                     Default: false



Given the Jacobian J of a genus 2 curve over the rationals, in 
the form y^2 = f(x) with integral coefficients, this computes a 
real bound on the absolute difference of the naive height on the 
Kummer surface of J and the canonical height on J.


The parameter Effort indicates how much effort should be put 
into finding a good bound.  The second value returned is a bound 
for mu_Infinity.  If the parameter Factor is true, then
the discriminant will be factored, and its prime divisors will be 
considered individually, usually resulting in an improvement of 
the bound.





NaiveHeight(P) : JacHypPt -> FldPrElt


NaiveHeight(P) : SrfKumPt -> FldPrElt



Returns the logarithmic naive height of the point P on a Jacobian J or 
its Kummer surface K, defined by the mappings J -> K -> ( P)^3, to projective space.





Height(P: Precision) : JacHypPt -> FldPrElt


Height(P: Precision) : SrfKumPt -> FldPrElt


CanonicalHeight(P: Precision) : JacHypPt  -> FldPrElt


CanonicalHeight(P: Precision) : SrfKumPt -> FldPrElt



    Precision: RngIntElt                Default: 0



Returns the canonical height of a point P on the Jacobian or 
Kummer surface of a genus 2 curve over the rationals.



HeightPairing(P, Q: Precision) : JacHypPt, JacHypPt -> FldPrElt



    Precision: RngIntElt                Default: 0



Returns the canonical height pairing of two rational points 
P and Q on the Jacobian of a genus 2 curve, defined by  
< P, Q > := (h(P + Q) - h(P) - h(Q))/2.  





HeightPairingMatrix(S: Precision) : [JacHypPt] -> AlgMat



    Precision: RngIntElt                Default: 0



Given a sequence S of points on the Jacobian J of a hyperelliptic
curve, this function returns the height pairing matrix of the sequence 
of points S.





Regulator(S: Precision) : [JacHypPt] -> FldPrElt



    Precision: RngIntElt                Default: 0



Given a sequence S of points on the Jacobian J of a hyperelliptic
curve, the function returns the determinant of the height pairing matrix 
of S.  The regulator is equal to zero when the points are dependent 
in the Mordell--Weil group, and otherwise is equal to the square of the
volume of the parallelotope spanned by the points in the subgroup of 
the free quotient of J(Q) generated by S.





ReducedBasis(S: Precision) : [JacHypPt] -> SeqEnum, AlgMatElt


ReducedBasis(S: Precision) : @JacHypPt@ -> SeqEnum, AlgMatElt


ReducedBasis(S: Precision) : JacHypPt -> SeqEnum, AlgMatElt



    Precision: RngIntElt                Default: 0



Given a sequence S of points on the Jacobian J of a hyperelliptic
curve, this function returns a reduced basis, with respect to 
the height pairing, for the subgroup of the free quotient of J(Q) 
generated by the given points on J.  The height pairing matrix of 
the sequence S is returned as a second value.





Example CrvHyp_HeightPairing (H86E12)

We compute a reduced basis for a set of points on the Jacobian J 
of a genus 2 curve C.  We first construct a collection of points 
P_1, P_2, P_3, and P_4 on the curve, then form points Q_1,
Q_2, and Q_3 on the Jacobian as degree zero linear combinations 
of the P_i.  We then compute a reduced basis, showing that the 
group generated by the points has rank 2.  The final nonzero 
determinant of the height pairing matrix proves the independence of 
the points.





> P<x> := PolynomialRing(Rationals());
> C := HyperellipticCurve(x^6+x^2+1);         
> J := Jacobian(C);
> Z := PointsAtInfinity(C);
> Z;
{@ (1 : -1 : 0), (1 : 1 : 0) @}
> P1 := Z[1]; 
> P2 := Z[2]; 
> P3 := C![1/2,9/8,1];
> P4 := C![-1/2,9/8,1];
> Q1 := J![P1,P2];
> Q2 := J![P1,P3];
> Q3 := J![P3,P4];
> B, M := ReducedBasis([Q1,Q2,Q3] : Precision := 12);   
> B;
[ (1, -x^3, 2), (x^2, 1, 2) ]
> M;
[0.2797317933688278287667056839      -5.005347463630776922E-17]
[     -5.005347463630776922E-17 0.9524462128097307697800077959]
> Determinant(M);
0.2664294871966142247655164984





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